Examples and Counterexamples最新文献

The change of basis matrix $M$ from shifted Legendre to Bernstein polynomials and $M^{-1}$ have applications in computer graphics. Algorithms use their properties to find the matrix elements efficiently. We give new functions for the elements of $M$ as a summation, and a complete hypergeometric function. We find that Gosper’s algorithm does not produce closed-form expressions for the elements of either $M$ or $M^{-1}$. Zeilberger’s algorithm produces four second-order recurrences for the elements of the matrices that enable them to be computed in $O\left(n\right)$ time, and for the derivation of closed-form functions by row and column for the elements. Two row recurrences are special cases of those found by Woźny (2013) who used a different method. We show that the recurrences for rows of $M$ and columns of $M^{-1}$ are equivalent. The recurrences for columns of $M$ and rows of $M^{-1}$ generate functions that are the Lagrange interpolation polynomials of their elements. These polynomials are equal to hypergeometric functions, which are solutions of the recurrences.

Two resolutions of the same design are said to be orthogonal when each parallel class of one resolution has at most one block in common with each parallel class of the other resolution. If a candelabra quadruple system has two mutually orthogonal resolutions, the design is called doubly resolvable candelabra quadruple system and denoted by DRCQS. In this paper, we obtain a DRCQS$(3^5:1)$ by computer search.

Any set $S$ of elements from an abelian group produces a graph with colored edges $G$(S), with its points the elements of $S$, and the edge between points $P$ and $Q$ assigned for its “color” the sum $P+Q$. Since any pair of identically colored edges is equivalent to an equation $P+Q=P^{\prime }+Q^{\prime }$, the geometric—combinatorial figure $G$(S) is thus equivalent to a system of linear equations. This article derives elementary properties of such “sum cographs”, including forced or forbidden configurations, and then catalogues the 54 possible sum cographs on up to 6 points. Larger sum cograph structures also exist: Points $\left\{P_i\right\}$ in $Z_m$ close up into a “Fibonacci cycle”–i.e. $P_0=1$, $P_1=k$, $P_{i+2}=P_i+P_{i+1}$ for all integers $i\ge 0$, and then $P_n=P_0$ and $P_{n+1}=P_1$–provided that $m=L_n$ is a Lucas prime, in which case actually $P_i=k^i$ for all $i\ge 0$.

The Sombor matrix of a graph $G$ with vertices $v_1,v_2,\dots ,v_n$ is defined as $A_{SO}\left(G\right)=\left[s_{ij}\right]$, where $s_{ij}=\sqrt{d_i^2+d_j^2}$ if $v_i$ is adjacent to $v_j$ and $s_{ij}=0$, otherwise, where $d_i$ is the degree of a vertex $v_i$. The Sombor energy of a graph is defined as the sum of the absolute values of the eigenvalues of the Sombor matrix. N. Ghanbari (Ghanbari, 2022) conjectured that there is no graph with integer valued Sombor energy. In this paper we give some class of graphs for which this conjecture holds. Further we conjecture that there is no regular graph with adjacency energy equal to $2k\sqrt{2}$ where $k$ is a positive integer.

In the present paper, we show an example of a solution for Dorodnitzyn’s gaseous boundary layer limit formula. Oleinik’s no back-flow condition ensures the existence and uniqueness of solutions for the Prandtl equations in a rectangular domain $R\subset R^2$. It also allowed us to find a limit formula for Dorodnitzyn’s stationary compressible boundary layer with constant total energy on a bounded convex domain in the plane $R^2$. Under the same assumption, we can give an approximate solution $u$ for the limit formula if $\left|u\right|<<<1$ such that: $u\left(z\right)\cong \delta \ast c\ast \left(z+\frac{6}{25}\cdot \frac{1}{2i_0}\cdot \frac{4U^2}{3}{z}^4\right)+o\left(z^5\right),$that corresponds to an approximate horizontal velocity component when a small parameter $ϵ$ given by the quotient of the maximum height of the domain divided by its length tends to zero. Here, $c>0$, $\delta$ is the boundary layer’s height in Dorodnitzyn’s coordinates, $U$ is the free-stream velocity at the upper boundary of the domain, and $T_0$ is the absolute surface temperature.

We provide a condition on the occurrences of variables in polynomials and show that it ensures associativity of univariate resultants in non-trivial cases. We give examples involving transformations and arithmetic with the zeros of polynomials. Associativity enables the composition of functions on the zeros of a polynomial by using resultants. The result is generalised to finite systems of polynomials.

In this paper, we prove that when the number of terms in the group determinant of order odd prime $p$ is divided by $p$, the remainder is 1. In addition, we give a table of the number of terms in $k$th power of the group determinant of the cyclic group of order $n$ for $n\le 10$ and $k\le 6$, and also give a table of one for every group of order at most 15. These tables raise some questions for us about the number of terms in the group determinants.

In this article, we present some rare counterexamples, which are related to (strong) persistence property and (nearly) normally torsion-freeness of monomial ideals. They may be useful for researchers in this field to construct the other counterexamples refuting some conjectures.

In this work we prove the Ulam–Hyers stability of the following equation $\left(E\right){\varphi }^{\prime \prime }\left(x\right)+(\gamma -1){\varphi }^{\prime }\left(x\right)-\gamma \varphi \left(x\right)=0,$where $\gamma$ is a real number. The main purpose is to find a solution $\varphi$ of (E) satisfying $|\varphi \left(x\right)-f\left(x\right)|⩽K\varepsilon$, where $K$ is Ulam–Hyers-Stability constant and $f$ is an exact solution of the associated inequality $|f^{\prime \prime }\left(x\right)+(\gamma -1)f^{\prime }\left(x\right)-\gamma f\left(x\right)|⩽\varepsilon ,$for any $\varepsilon >0$.

The radial map $u\left(x\right)=\frac{x}{\Vert x\Vert }$ is a well-known example of a harmonic map into the spheres with a point singularity at $x=0$. In our previous paper (Misawa and Nakauchi, 2022) we give two examples of harmonic maps into the standard spheres of higher dimension with a singularity of a polynomial of $\frac{x_1}{\Vert x\Vert },\cdots \frac{x_m}{\Vert x\Vert }$ of degree 2 and degree 3 respectively. In Fujioka (2020) uses our arguments to give an example of a harmonic map into the sphere with a singularity of a polynomial of degree 4. In this paper we give a family of examples of harmonic maps with a point singularity of a polynomial of higher degree.